RD Sharma solution class 8 chapter 4 Cube and Cube roots Exercise 4.5

Exercise 4.5

Page-4.36

Question 1:

Making use of the cube root table, find the cube roots 7

Answer 1:

Because 7 lies between 1 and 100, we will look at the row containing 7 in the column of x.

By the cube root table, we have:

$\sqrt[3]{7}=1.913$

Thus, the answer is 1.913.

Question 2:

Making use of the cube root table, find the cube root
70

Answer 2:

Because 70 lies between 1 and 100, we will look at the row containing 70 in the column of x.

By the cube root table, we have:

$\sqrt[3]{70}=4.121$

Question 3:

Making use of the cube root table, find the cube root
700

Answer 3:

We have:

$700=70\times 10$

$\therefore$ Cube root of 700 will be in the column of $\sqrt[3]{10x}$ against 70.

By the cube root table, we have:

$\sqrt[3]{700}=8.879$

Thus, the answer is 8.879.

Question 4:

Making use of the cube root table, find the cube root
7000

Answer 4:

We have:

$7000=70\times 100$

$\therefore$  $\sqrt[3]{7000}=\sqrt[3]{7\times 1000}=\sqrt[3]{7}\times \sqrt[3]{1000}$ 

By the cube root table, we have: 

$\sqrt[3]{7}=1.913 \text{and} \sqrt[3]{1000}=10$

$\therefore$ $\sqrt[3]{7000}=\sqrt[3]{7}\times \sqrt[3]{1000}=1.913\times 10=19.13$

Question 5:

Making use of the cube root table, find the cube root
1100

Answer 5:

We have:

$1100=11\times 100$

$\therefore$  $\sqrt[3]{1100}=\sqrt[3]{11\times 100}=\sqrt[3]{11}\times \sqrt[3]{100}$ 

By the cube root table, we have: 

$\sqrt[3]{11}=2.224 \text{and} \sqrt[3]{100}=4.642$

$\therefore$ $\sqrt[3]{1100}=\sqrt[3]{11}\times \sqrt[3]{100}=2.224\times 4.642=10.323 (\mathrm{Up} \mathrm{to} \mathrm{three} \mathrm{decimal} \mathrm{places})$

Thus, the answer is 10.323.

Question 6:

Making use of the cube root table, find the cube root
780

Answer 6:

We have:

$780=78\times 10$

$\therefore$ Cube root of 780 would be in the column of $\sqrt[3]{10x}$ against 78.

By the cube root table, we have:

$\sqrt[3]{780}=9.205$

Thus, the answer is 9.205.

Question 7:

Making use of the cube root table, find the cube root
7800

Answer 7:

We have:

$7800=78\times 100$

$\therefore$ $\sqrt[3]{7800}=\sqrt[3]{78\times 100}=\sqrt[3]{78}\times \sqrt[3]{100}$ 

By the cube root table, we have:
 
$\sqrt[3]{78}=4.273 \text{and} \sqrt[3]{100}=4.642$

$\sqrt[3]{7800}=\sqrt[3]{78}\times \sqrt[3]{100}=4.273\times 4.642=19.835 (upto three decimal places)$

Thus, the answer is 19.835

Question 8:

Making use of the cube root table, find the cube root
1346

Answer 8:

By prime factorisation, we have:

$1346=2\times 673\Rightarrow \sqrt[3]{1346}=\sqrt[3]{2}\times \sqrt[3]{673}$

Also

$670<673<680\Rightarrow \sqrt[3]{670}<\sqrt[3]{673}<\sqrt[3]{680}$

From the cube root table, we have:

$\sqrt[3]{670}=8.750 \text{and} \sqrt[3]{680}=8.794$ 

For the difference (680$-$670), i.e., 10, the difference in the values

$=8.794-8.750=0.044$

$\therefore$ For the difference of (673$-$670), i.e., 3, the difference in the values

$=\frac{0.044}{10}\times 3=0.0132=0.013$ (upto three decimal places)

$\therefore$ $\sqrt[3]{673}=8.750+0.013=8.763$

Now

$\sqrt[3]{1346}=\sqrt[3]{2}\times \sqrt[3]{673}=1.260\times 8.763=11.041$ (upto three decimal places)

Thus, the answer is 11.041.

Question 9:

Making use of the cube root table, find the cube root
250

Answer 9:

We have:

$250=25\times 100$

$\therefore$ Cube root of 250 would be in the column of $\sqrt[3]{10x}$ against 25.

By the cube root table, we have:

$\sqrt[3]{250}=6.3$

Thus, the required cube root is 6.3.

Question 10:

Making use of the cube root table, find the cube root
5112

Answer 10:

By prime factorisation, we have:

$5112=2^3\times 3^2\times 71\Rightarrow \sqrt[3]{5112}=2\times \sqrt[3]{9}\times \sqrt[3]{71}$

By the cube root table, we have:
 
$\sqrt[3]{9}=2.080 and \sqrt[3]{71}=4.141$

$\therefore$ $\sqrt[3]{5112}=2\times \sqrt[3]{9}\times \sqrt[3]{71}=2\times 2.080\times 4.141=17.227$ (upto three decimal places)

Thus, the required cube root is 17.227.

Question 11:

Making use of the cube root table, find the cube root
9800

Answer 11:

We have:

$9800=98\times 100$

$\therefore$  $\sqrt[3]{9800}=\sqrt[3]{98\times 100}=\sqrt[3]{98}\times \sqrt[3]{100}$ 

By cube root table, we have: 

$\sqrt[3]{98}=4.610 \text{and} \sqrt[3]{100}=4.642$

$\therefore$ $\sqrt[3]{9800}=\sqrt[3]{98}\times \sqrt[3]{100}=4.610\times 4.642=21.40$ (upto three decimal places)

Thus, the required cube root is 21.40.

Question 12:

Making use of the cube root table, find the cube root
732

Answer 12:

We have:

$730<732<740\Rightarrow \sqrt[3]{730}<\sqrt[3]{732}<\sqrt[3]{740}$

From cube root table, we have:

$\sqrt[3]{730}=9.004 \text{and} \sqrt[3]{740}=9.045$ 

For the difference (740$-$730), i.e., 10, the difference in values

$=9.045-9.004=0.041$

$\therefore$ For the difference of (732$-$730), i.e., 2, the difference in values

$=\frac{0.041}{10}\times 2=0.0082$

 $\therefore$ $\sqrt[3]{732}=9.004+0.008=9.012$

Question 13:

Making use of the cube root table, find the cube root
7342

Answer 13:

We have:

$7300<7342<7400\Rightarrow \sqrt[3]{7000}<\sqrt[3]{7342}<\sqrt[3]{7400}$

From the cube root table, we have: 

$\sqrt[3]{7300}=19.39 \text{and} \sqrt[3]{7400}=19.48$ 

For the difference (7400$-$7300), i.e., 100, the difference in values

$=19.48-19.39=0.09$

$\therefore$ For the difference of (7342$-$7300), i.e., 42, the difference in the values

$=\frac{0.09}{100}\times 42=0.0378=0.037$

$\therefore$ $\sqrt[3]{7342}=19.39+0.037=19.427$

Question 14:

Making use of the cube root table, find the cube root
133100

Answer 14:

We have:

$133100=1331\times 100\Rightarrow \sqrt[3]{133100}=\sqrt[3]{1331\times 100}=11\times \sqrt[3]{100}$

By cube root table, we have: 

$\sqrt[3]{100}=4.642$

$\therefore$ $\sqrt[3]{133100}=11\times \sqrt[3]{100}=11\times 4.642=51.062$

Question 15:

Making use of the cube root table, find the cube root
37800

Answer 15:

We have:

$37800=2^3\times 3^3\times 175\Rightarrow \sqrt[3]{37800}=\sqrt[3]{2^3\times 3^3\times 175}=6\times \sqrt[3]{175}$

Also

$170<175<180\Rightarrow \sqrt[3]{170}<\sqrt[3]{175}<\sqrt[3]{180}$

From cube root table, we have: 

$\sqrt[3]{170}=5.540 \text{and} \sqrt[3]{180}=5.646$

For the difference (180$-$170), i.e., 10, the difference in values

$=5.646-5.540=0.106$

$\therefore$ For the difference of (175$-$170), i.e., 5, the difference in values

$=\frac{0.106}{10}\times 5=0.053$

$\therefore$ $\sqrt[3]{175}=5.540+0.053=5.593$

Now

$37800=6\times \sqrt[3]{175}=6\times 5.593=33.558$

Thus, the required cube root is 33.558.

Question 16:

Making use of the cube root table, find the cube root
0.27

Answer 16:

The number 0.27 can be written as $\frac{27}{100}$.

Now

$\sqrt[3]{0.27}=\sqrt[3]{\frac{27}{100}}=\frac{\sqrt[3]{27}}{\sqrt[3]{100}}=\frac{3}{\sqrt[3]{100}}$

By cube root table, we have: 

$\sqrt[3]{100}=4.642$

$\therefore$ $\sqrt[3]{0.27}=\frac{3}{\sqrt[3]{100}}=\frac{3}{4.642}=0.646$

Thus, the required cube root is 0.646.

Question 17:

Making use of the cube root table, find the cube root
8.6

Answer 17:

The number 8.6 can be written as $\frac{86}{10}$.

Now

$\sqrt[3]{8.6}=\sqrt[3]{\frac{86}{10}}=\frac{\sqrt[3]{86}}{\sqrt[3]{10}}$

By cube root table, we have:
 
$\sqrt[3]{86}=4.414 \text{and} \sqrt[3]{10}=2.154$

$\therefore$ $\sqrt[3]{8.6}=\frac{\sqrt[3]{86}}{\sqrt[3]{10}}=\frac{4.414}{2.154}=2.049$

Thus, the required cube root is 2.049.

Question 18:

Making use of the cube root table, find the cube root
0.86

Answer 18:

The number 0.86 could be written as $\frac{86}{100}$.

Now

$\sqrt[3]{0.86}=\sqrt[3]{\frac{86}{100}}=\frac{\sqrt[3]{86}}{\sqrt[3]{100}}$

By cube root table, we have: 

$\sqrt[3]{86}=4.414 \text{and} \sqrt[3]{100}=4.642$

$\therefore$ $\sqrt[3]{0.86}=\frac{\sqrt[3]{86}}{\sqrt[3]{100}}=\frac{4.414}{4.642}=0.951$ (upto three decimal places)

Thus, the required cube root is 0.951.

Question 19:

Making use of the cube root table, find the cube root
8.65

Answer 19:

The number 8.65 could be written as $\frac{865}{100}$.

Now

$\sqrt[3]{8.65}=\sqrt[3]{\frac{865}{100}}=\frac{\sqrt[3]{865}}{\sqrt[3]{100}}$

Also

$860<865<870\Rightarrow \sqrt[3]{860}<\sqrt[3]{865}<\sqrt[3]{870}$

From the cube root table, we have: 

$\sqrt[3]{860}=9.510 \text{and} \sqrt[3]{870}=9.546$ 

For the difference (870$-$860), i.e., 10, the difference in values

$=9.546-9.510=0.036$

$\therefore$ For the difference of (865$-$860), i.e., 5, the difference in values

$=\frac{0.036}{10}\times 5=0.018$  (upto three decimal places)

$\therefore$ $\sqrt[3]{865}=9.510+0.018=9.528$ (upto three decimal places)

From the cube root table, we also have:

$\sqrt[3]{100}=4.642$

$\therefore$ $\sqrt[3]{8.65}=\frac{\sqrt[3]{865}}{\sqrt[3]{100}}=\frac{9.528}{4.642}=2.053$ (upto three decimal places)

Thus, the required cube root is 2.053.

Question 20:

Making use of the cube root table, find the cube root
7532

Answer 20:

We have:

$7500<7532<7600\Rightarrow \sqrt[3]{7500}<\sqrt[3]{7532}<\sqrt[3]{7600}$

From the cube root table, we have: 

$\sqrt[3]{7500}=19.57 \text{and} \sqrt[3]{7600}=19.66$ 

For the difference (7600$-$7500), i.e., 100, the difference in values

$=19.66-19.57=0.09$

$\therefore$ For the difference of (7532$-$7500), i.e., 32, the difference in values

$=\frac{0.09}{100}\times 32=0.0288=0.029$ (up to three decimal places)

$\therefore$ $\sqrt[3]{7532}=19.57+0.029=19.599$

Question 21:

Making use of the cube root table, find the cube root
833

Answer 21:

We have:

$830<833<840\Rightarrow \sqrt[3]{830}<\sqrt[3]{833}<\sqrt[3]{840}$

From the cube root table, we have: 

$\sqrt[3]{830}=9.398 \text{and} \sqrt[3]{840}=9.435$

For the difference (840$-$830), i.e., 10, the difference in values

$=9.435-9.398=0.037$

$\therefore$ For the difference (833$-$830), i.e., 3, the difference in values

$=\frac{0.037}{10}\times 3=0.0111=0.011$ (upto three decimal places)

$\therefore$ $\sqrt[3]{833}=9.398+0.011=9.409$

Question 22:

Making use of the cube root table, find the cube root
34.2

Answer 22:

The number 34.2 could be written as $\frac{342}{10}$.

Now

$\sqrt[3]{34.2}=\sqrt[3]{\frac{342}{10}}=\frac{\sqrt[3]{342}}{\sqrt[3]{10}}$

Also
 
$340<342<350\Rightarrow \sqrt[3]{340}<\sqrt[3]{342}<\sqrt[3]{350}$

From the cube root table, we have: 

$\sqrt[3]{340}=6.980 \text{and} \sqrt[3]{350}=7.047$ 

For the difference (350$-$340), i.e., 10, the difference in values

$=7.047-6.980=0.067$.

$\therefore$ For the difference (342$-$340), i.e., 2, the difference in values

$=\frac{0.067}{10}\times 2=0.013$  (upto three decimal places)

$\therefore$ $\sqrt[3]{342}=6.980+0.0134=6.993$ (upto three decimal places)

From the cube root table, we also have: 

$\sqrt[3]{10}=2.154$

$\therefore$ $\sqrt[3]{34.2}=\frac{\sqrt[3]{342}}{\sqrt[3]{10}}=\frac{6.993}{2.154}=3.246$

Thus, the required cube root is 3.246.

Question 23:

What is the length of the side of a cube whose volume is 275 cm³. Make use of the table for the cube root.

Answer 23:

Volume of a cube is given by: 

$V=a^3$, where a = side of the cube 

$\therefore$ Side of a cube = $a=\sqrt[3]{V}$

If the volume of a cube is 275 cm³, the side of the cube will be $\sqrt[3]{275}$.

We have:

$270<275<280\Rightarrow \sqrt[3]{270}<\sqrt[3]{275}<\sqrt[3]{280}$

From the cube root table, we have: 

$\sqrt[3]{270}=6.463 \text{and} \sqrt[3]{280}=6.542$.

For the difference (280$-$270), i.e., 10, the difference in values

$=6.542-6.463=0.079$

$\therefore$ For the difference (275$-$270), i.e., 5, the difference in values

$=\frac{0.079}{10}\times 5=0.0395 \simeq 0.04$ (upto three decimal places)

$\therefore$ $\sqrt[3]{275}=6.463+0.04=6.503$ (upto three decimal places)

Thus, the length of the side of the cube is 6.503 cm.